| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s − 11-s − 12-s + 13-s − 4·14-s + 16-s − 18-s + 2·19-s − 4·21-s + 22-s + 24-s − 5·25-s − 26-s − 27-s + 4·28-s + 6·29-s − 8·31-s − 32-s + 33-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.213·22-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247962 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247962 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304518663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304518663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.78477238845264, −12.13475700205725, −11.81478054659578, −11.42433600648025, −10.98217786296855, −10.65503661542600, −10.11351110357254, −9.710298302826952, −9.057709397973408, −8.595209298813628, −8.210968364641214, −7.677294809218484, −7.286778569307076, −6.917947268210062, −6.030655784014180, −5.763945820232929, −5.257246191428325, −4.695052504185405, −4.205350076590459, −3.608999999100103, −2.803863561987119, −2.220834904937968, −1.536378916543885, −1.244263528255955, −0.3788705991103740,
0.3788705991103740, 1.244263528255955, 1.536378916543885, 2.220834904937968, 2.803863561987119, 3.608999999100103, 4.205350076590459, 4.695052504185405, 5.257246191428325, 5.763945820232929, 6.030655784014180, 6.917947268210062, 7.286778569307076, 7.677294809218484, 8.210968364641214, 8.595209298813628, 9.057709397973408, 9.710298302826952, 10.11351110357254, 10.65503661542600, 10.98217786296855, 11.42433600648025, 11.81478054659578, 12.13475700205725, 12.78477238845264
